Hardcover ISBN:  9780821875810 
Product Code:  SURV/181 
367 pp 
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Electronic ISBN:  9780821887905 
Product Code:  SURV/181.E 
367 pp 
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Book DetailsMathematical Surveys and MonographsVolume: 181; 2012MSC: Primary 13; 16;
This book is a comprehensive treatment of the representation theory of maximal CohenMacaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras.
Two introductory chapters treat the KrullRemakSchmidt Theorem on uniqueness of directsum decompositions and its failure for modules over local rings. Chapters 3–10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material—ADE/simple singularities, the double branched cover, AuslanderReiten theory, and the BrauerThrall conjectures—is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and BurbanDrozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal CohenMacaulay modules, AuslanderBuchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.ReadershipResearch mathematicians interested in algebra, in particular, theory of rings and modules.

Table of Contents

Chapters

1. The KrullRemakSchmidt theorem

2. Semigroups of modules

3. Dimension zero

4. Dimension one

5. Invariant theory

6. Kleinian singularities and finite CM type

7. Isolated singularities and dimension two

8. The double branched cover

9. Hypersurfaces with finite CM type

10. Ascent and descent

11. AuslanderBuchweitz theory

12. Totally reflexive modules

13. AuslanderReiten theory

14. Countable CohenMacaulay type

15. The BrauerThrall conjectures

16. Finite CM type in higher dimensions

17. Bounded CM type

Appendix A. Basics and background

Appendix B. Ramification theory


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This book is a comprehensive treatment of the representation theory of maximal CohenMacaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras.
Two introductory chapters treat the KrullRemakSchmidt Theorem on uniqueness of directsum decompositions and its failure for modules over local rings. Chapters 3–10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material—ADE/simple singularities, the double branched cover, AuslanderReiten theory, and the BrauerThrall conjectures—is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and BurbanDrozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal CohenMacaulay modules, AuslanderBuchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.
Research mathematicians interested in algebra, in particular, theory of rings and modules.

Chapters

1. The KrullRemakSchmidt theorem

2. Semigroups of modules

3. Dimension zero

4. Dimension one

5. Invariant theory

6. Kleinian singularities and finite CM type

7. Isolated singularities and dimension two

8. The double branched cover

9. Hypersurfaces with finite CM type

10. Ascent and descent

11. AuslanderBuchweitz theory

12. Totally reflexive modules

13. AuslanderReiten theory

14. Countable CohenMacaulay type

15. The BrauerThrall conjectures

16. Finite CM type in higher dimensions

17. Bounded CM type

Appendix A. Basics and background

Appendix B. Ramification theory